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**Math Olympiad Group Problems**

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**The large square has an area of 100 square meters**

The large square has an area of 100 square meters. The vertices of the other square lie at the midpoints of the sides of the larger square. What is the area of the smaller square?

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50 square meters

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**What’s the missing number?**

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39

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Solve the riddle: I am a 2-digit prime number I am greater than 20 and less than 60 The sum of my digits is 10 Who am I?

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37

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**Greg tosses 4 coins in the air and they come up 3 heads and 1 tail**

Greg tosses 4 coins in the air and they come up 3 heads and 1 tail. Express the probability of this happening as a reduced fraction.

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**One angle of a triangle has a measurement of 70 degrees**

One angle of a triangle has a measurement of 70 degrees. The ratio of the other two angles of the triangle is 5:6. What are the measurements of the other two angles?

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50 and 60 degrees

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King Kong climbed to the top of the 1250 foot tall Empire State Building. He would climb 150 feet, but then slip down 25 feet. How many feet total did King Kong climb up during his journey to get to the top?

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1475 feet

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**The odometer on a car reads 42322 km**

The odometer on a car reads km. In the next 200 km how often will the odometer show a number in which exactly three of the digits are the same?

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22

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A set of 5 whole numbers all more than 0 but less than 10 has a mean of 5, a mode of 2, and a median less than the mean. What is the set of numbers?

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{2, 2, 4, 8, 9}

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**A group of scientists is duplicating cells for a science experiment**

A group of scientists is duplicating cells for a science experiment. They observe that during each 10-minute period, one-quarter of the cells present died and the rest of the cells all doubled. If there were 32 cells initially, how many cells were there after 40 minutes?

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162 cells

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**How many integers satisfy:**

(n – 4)(n + 6) < 0?

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9

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**A baseball team won 20% of its games and lost 12 games**

A baseball team won 20% of its games and lost 12 games. How many games did the team play?

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15

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**The sum of the areas of the two congruent circles shown is 72 ft. 2**

The sum of the areas of the two congruent circles shown is 72 ft.2. What is the area of rectangle ABCD?

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288 sq. ft

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**Square ABCD has a perimeter of 20 centimeters**

Square ABCD has a perimeter of 20 centimeters. If a circle is inscribed in the square, as shown, what is the area of the circle?

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or

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In an Olympic event, the winner of a triathlon swam 1 hour 42 minutes, then biked for 6 hours 59 minutes, then ran for 6 hours 13 minutes. If he started at 9:00 a.m., when did he finish?

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11:54 p.m.

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**A genius reads 100 words backwards per second**

A genius reads 100 words backwards per second. How long does it take her to read 31,000 words backward?

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5 minutes and 10 seconds

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If each small circle has a diameter of 6 cm, and if the two small circles touch at the center of the large circle, what is the area of the shaded region?

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or 56.55

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